(* Author: Tobias Nipkow *)
theory ACom
imports Com
begin
subsection "Annotated Commands"
datatype 'a acom =
SKIP 'a ("SKIP {_}" 61) |
Assign vname aexp 'a ("(_ ::= _/ {_})" [1000, 61, 0] 61) |
Seq "('a acom)" "('a acom)" ("_;;//_" [60, 61] 60) |
If bexp 'a "('a acom)" 'a "('a acom)" 'a
("(IF _/ THEN ({_}/ _)/ ELSE ({_}/ _)//{_})" [0, 0, 0, 61, 0, 0] 61) |
While 'a bexp 'a "('a acom)" 'a
("({_}//WHILE _//DO ({_}//_)//{_})" [0, 0, 0, 61, 0] 61)
text_raw{*\snip{stripdef}{1}{1}{% *}
fun strip :: "'a acom \<Rightarrow> com" where
"strip (SKIP {P}) = com.SKIP" |
"strip (x ::= e {P}) = x ::= e" |
"strip (C\<^sub>1;;C\<^sub>2) = strip C\<^sub>1;; strip C\<^sub>2" |
"strip (IF b THEN {P\<^sub>1} C\<^sub>1 ELSE {P\<^sub>2} C\<^sub>2 {P}) =
IF b THEN strip C\<^sub>1 ELSE strip C\<^sub>2" |
"strip ({I} WHILE b DO {P} C {Q}) = WHILE b DO strip C"
text_raw{*}%endsnip*}
text_raw{*\snip{asizedef}{1}{1}{% *}
fun asize :: "com \<Rightarrow> nat" where
"asize com.SKIP = 1" |
"asize (x ::= e) = 1" |
"asize (C\<^sub>1;;C\<^sub>2) = asize C\<^sub>1 + asize C\<^sub>2" |
"asize (IF b THEN C\<^sub>1 ELSE C\<^sub>2) = asize C\<^sub>1 + asize C\<^sub>2 + 3" |
"asize (WHILE b DO C) = asize C + 3"
text_raw{*}%endsnip*}
text_raw{*\snip{annotatedef}{1}{1}{% *}
definition shift :: "(nat \<Rightarrow> 'a) \<Rightarrow> nat \<Rightarrow> nat \<Rightarrow> 'a" where
"shift f n = (\<lambda>p. f(p+n))"
fun annotate :: "(nat \<Rightarrow> 'a) \<Rightarrow> com \<Rightarrow> 'a acom" where
"annotate f com.SKIP = SKIP {f 0}" |
"annotate f (x ::= e) = x ::= e {f 0}" |
"annotate f (c\<^sub>1;;c\<^sub>2) = annotate f c\<^sub>1;; annotate (shift f (asize c\<^sub>1)) c\<^sub>2" |
"annotate f (IF b THEN c\<^sub>1 ELSE c\<^sub>2) =
IF b THEN {f 0} annotate (shift f 1) c\<^sub>1
ELSE {f(asize c\<^sub>1 + 1)} annotate (shift f (asize c\<^sub>1 + 2)) c\<^sub>2
{f(asize c\<^sub>1 + asize c\<^sub>2 + 2)}" |
"annotate f (WHILE b DO c) =
{f 0} WHILE b DO {f 1} annotate (shift f 2) c {f(asize c + 2)}"
text_raw{*}%endsnip*}
text_raw{*\snip{annosdef}{1}{1}{% *}
fun annos :: "'a acom \<Rightarrow> 'a list" where
"annos (SKIP {P}) = [P]" |
"annos (x ::= e {P}) = [P]" |
"annos (C\<^sub>1;;C\<^sub>2) = annos C\<^sub>1 @ annos C\<^sub>2" |
"annos (IF b THEN {P\<^sub>1} C\<^sub>1 ELSE {P\<^sub>2} C\<^sub>2 {Q}) =
P\<^sub>1 # annos C\<^sub>1 @ P\<^sub>2 # annos C\<^sub>2 @ [Q]" |
"annos ({I} WHILE b DO {P} C {Q}) = I # P # annos C @ [Q]"
text_raw{*}%endsnip*}
definition anno :: "'a acom \<Rightarrow> nat \<Rightarrow> 'a" where
"anno C p = annos C ! p"
definition post :: "'a acom \<Rightarrow>'a" where
"post C = last(annos C)"
text_raw{*\snip{mapacomdef}{1}{2}{% *}
fun map_acom :: "('a \<Rightarrow> 'b) \<Rightarrow> 'a acom \<Rightarrow> 'b acom" where
"map_acom f (SKIP {P}) = SKIP {f P}" |
"map_acom f (x ::= e {P}) = x ::= e {f P}" |
"map_acom f (C\<^sub>1;;C\<^sub>2) = map_acom f C\<^sub>1;; map_acom f C\<^sub>2" |
"map_acom f (IF b THEN {P\<^sub>1} C\<^sub>1 ELSE {P\<^sub>2} C\<^sub>2 {Q}) =
IF b THEN {f P\<^sub>1} map_acom f C\<^sub>1 ELSE {f P\<^sub>2} map_acom f C\<^sub>2
{f Q}" |
"map_acom f ({I} WHILE b DO {P} C {Q}) =
{f I} WHILE b DO {f P} map_acom f C {f Q}"
text_raw{*}%endsnip*}
lemma annos_ne: "annos C \<noteq> []"
by(induction C) auto
lemma strip_annotate[simp]: "strip(annotate f c) = c"
by(induction c arbitrary: f) auto
lemma length_annos_annotate[simp]: "length (annos (annotate f c)) = asize c"
by(induction c arbitrary: f) auto
lemma size_annos: "size(annos C) = asize(strip C)"
by(induction C)(auto)
lemma size_annos_same: "strip C1 = strip C2 \<Longrightarrow> size(annos C1) = size(annos C2)"
apply(induct C2 arbitrary: C1)
apply(case_tac C1, simp_all)+
done
lemmas size_annos_same2 = eqTrueI[OF size_annos_same]
lemma anno_annotate[simp]: "p < asize c \<Longrightarrow> anno (annotate f c) p = f p"
apply(induction c arbitrary: f p)
apply (auto simp: anno_def nth_append nth_Cons numeral_eq_Suc shift_def
split: nat.split)
apply (metis add_Suc_right add_diff_inverse nat_add_commute)
apply(rule_tac f=f in arg_cong)
apply arith
apply (metis less_Suc_eq)
done
lemma eq_acom_iff_strip_annos:
"C1 = C2 \<longleftrightarrow> strip C1 = strip C2 \<and> annos C1 = annos C2"
apply(induction C1 arbitrary: C2)
apply(case_tac C2, auto simp: size_annos_same2)+
done
lemma eq_acom_iff_strip_anno:
"C1=C2 \<longleftrightarrow> strip C1 = strip C2 \<and> (\<forall>p<size(annos C1). anno C1 p = anno C2 p)"
by(auto simp add: eq_acom_iff_strip_annos anno_def
list_eq_iff_nth_eq size_annos_same2)
lemma post_map_acom[simp]: "post(map_acom f C) = f(post C)"
by (induction C) (auto simp: post_def last_append annos_ne)
lemma strip_map_acom[simp]: "strip (map_acom f C) = strip C"
by (induction C) auto
lemma anno_map_acom: "p < size(annos C) \<Longrightarrow> anno (map_acom f C) p = f(anno C p)"
apply(induction C arbitrary: p)
apply(auto simp: anno_def nth_append nth_Cons' size_annos)
done
lemma strip_eq_SKIP:
"strip C = com.SKIP \<longleftrightarrow> (EX P. C = SKIP {P})"
by (cases C) simp_all
lemma strip_eq_Assign:
"strip C = x::=e \<longleftrightarrow> (EX P. C = x::=e {P})"
by (cases C) simp_all
lemma strip_eq_Seq:
"strip C = c1;;c2 \<longleftrightarrow> (EX C1 C2. C = C1;;C2 & strip C1 = c1 & strip C2 = c2)"
by (cases C) simp_all
lemma strip_eq_If:
"strip C = IF b THEN c1 ELSE c2 \<longleftrightarrow>
(EX P1 P2 C1 C2 Q. C = IF b THEN {P1} C1 ELSE {P2} C2 {Q} & strip C1 = c1 & strip C2 = c2)"
by (cases C) simp_all
lemma strip_eq_While:
"strip C = WHILE b DO c1 \<longleftrightarrow>
(EX I P C1 Q. C = {I} WHILE b DO {P} C1 {Q} & strip C1 = c1)"
by (cases C) simp_all
lemma [simp]: "shift (\<lambda>p. a) n = (\<lambda>p. a)"
by(simp add:shift_def)
lemma set_annos_anno[simp]: "set (annos (annotate (\<lambda>p. a) c)) = {a}"
by(induction c) simp_all
lemma post_in_annos: "post C \<in> set(annos C)"
by(auto simp: post_def annos_ne)
lemma post_anno_asize: "post C = anno C (size(annos C) - 1)"
by(simp add: post_def last_conv_nth[OF annos_ne] anno_def)
end